Finite Group Databasebeta

← #9348

Subgroups — #9348

Order 256.

235 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3094C211yesmaximal, characteristic
2#2924C211yesmaximal, characteristic, complemented
3#1825C211yesmaximal, characteristic, complemented
4#1519C211yesmaximal, characteristic
5#1471C211yesmaximal, characteristic
6#1305C211yesmaximal, characteristic, complemented
7#1298C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×#117C2211yescharacteristic
10#482—21no—
11#482—21no—
12#481—21no—
13#480—21no—
14#433—22no—
15#432—21no—
16#431—21no—
17#428C411yescharacteristic, complemented
18#428C411yescharacteristic, complemented
19#424—21no—
20#418C411yescharacteristic, complemented
21#418C411yescharacteristic, complemented
22#407C2211yescharacteristic
23#407C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26#405—21no—
27#380C2211yescharacteristic
28#340—21no—
29#340—21no—
30#327—21no—
31#327—21no—
32#323—22no—
33C22×D4—21no—
34C22×D4C2×C411yescharacteristic
35C23×C4C2311yescharacteristic
36C2×M16—21no—
37C2×M16C2×C411yescharacteristic
38C2×M16C2×C411yescharacteristic
39C22×C8—21no—
40C22×C8—21no—
41#121—21no—
42#121—21no—
43#121—21no—
44#121—21no—
45#117—21no—
46#117—21no—
47#117C2×C411yescharacteristic
48#117C2×C411yescharacteristic
49#117D411yescharacteristic
50#117D411yescharacteristic
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2×C411yescharacteristic
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C42—21no—
59#107—21no—
60#107—21no—
61#106—21no—
62#106—21no—
63C4⋊C8—22no—
64C4⋊C8—22no—
65#102—21no—
66#102—21no—
67#102—21no—
68#102—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8D412yes—
76C22⋊C8D411yescharacteristic
77C22⋊C8D411yescharacteristic
78#97—22no—
79C4×C8—22no—
80#95—21no—
81C24—21no—
82C24#4111yescharacteristic
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—81no—
90C2×D4—81no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4C22×C411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4#4111yescharacteristic
103M16—41no—
104M16—41no—
105M16—41no—
106M16—41no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—22no—
111C2×C8—21no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4C22⋊C411yescharacteristic
124C4⋊C4C22⋊C411yescharacteristic
125C4⋊C4C22⋊C411yescharacteristic
126C4⋊C4C22⋊C411yescharacteristic
127C22⋊C4—21no—
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—21no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—81no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—21no—
138C23—21no—
139C23—21no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—41no—
144C23—81no—
145C23—81no—
146C23C4×D411yescharacteristic
147C23#12111yescharacteristic
148C23#12311yescharacteristic
149D4—41no—
150D4—41no—
151D4—81no—
152D4—82no—
153D4—81no—
154D4—81no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4C2×C22⋊C411yescharacteristic
177C2×C4C4×D411yescharacteristic
178C2×C4#12011yescharacteristic
179C2×C4#12111yescharacteristic
180C8—41no—
181C8—41no—
182C8—42no—
183C8—41no—
184C8—41no—
185C8—81no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22—162no—
200C22#38611yescharacteristic
201C22#47111yescharacteristic
202C22#47211yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—41no—
207C4—41no—
208C4—41no—
209C4—41no—
210C4—81no—
211C4—161no—
212C4#41011yescharacteristic
213C4#41011yescharacteristic
214C2—21no—
215C2—21no—
216C2—41no—
217C2—41no—
218C2—161no—
219C2#188111yescharacteristic
220C2#189911yescharacteristic
221C2#190711yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2924C2semidirect
3. #1825C2semidirect
6. #1305C2semidirect
7. #1298C2semidirect
17. #428C4semidirect
18. #428C4semidirect
20. #418C4semidirect
21. #418C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#30942
#29242
#18252
#15192
#14712
#13052
#12982